Chapter 9 Error-Correcting Constrained Coding

Chapter ErrorCorrecting Constrained Co ding In this chapter we consider co des that have combined errorcorrection and constrained prop erties We b egin with a discussion of error mechanisms in recording systems and the corresp onding error typ es observed We then discuss asso ciated metrics imp osed on con strained systemsprimarily the Hamming Lee and Euclidean metricsand we survey the literature on b ounds and co de constructions In addition we consider two imp ortant classes of combined errorcorrectionconstrained co des sp ectral null co des and forbidden list co des Errormechanisms in recording channels Magnetic recording systems using p eak detection as describ ed in Chapter of this chapter are sub ject to three predominant typ es of errors at the peak detector output The most frequently observed error is referred to as a bitshift error where a pair of recorded symbols is detected as a left bitshift or the pair is detected as a right bitshift Another commonly o ccurring error typ e is called a dropout error or sometimes a missingbit error where a recorded symbol is detected as a Less frequently a dropin error or extrabit error results in the detection of a recorded as a It is convenient to refer to the dropin and dropout errors as substitution errors Hammingmetric constrained co des are most p ertinent in recording channels that b ehave which dropin and dropout errors o ccur with equal like a binary symmetric channel in probability However there are alternative mo dels of interest that suggest the use of co des designed with other criteria in mind beyond optimization of minimum Hamming distance Among these mo dels the two that have received the most attention are the asymmetric channelwhere only dropin errors or dropout errors but not both are encountered and the bitshift channelwhere a symbol is allowed to shift p osition by up to a presp ecied umber of p ositions n CHAPTER ERRORCORRECTING CONSTRAINED CODING Another error typ e we will consider is a synchronization error resulting in an insertion or deletion of a symbol in the detected symbol sequence In practical digital recording systems on disks and tap e this typ e of error can have catastrophic consequences with re gard to recovery of information that follows the synchronization loss As a result recording devices use synchronization and clo ck generation techniques in conjunction with co de con the G constraint in straints such as the k constraint in RLL co des for p eak detection and PRML GI co des to eectively preclude such errors Nevertheless RLLconstrained synchronizationerrorcorrecting co des have some intrinsic co dingtheoretic interest and we will discuss them b elow Co des capable of correcting more than one insertion and deletion error may also b e used to protect against bitshift errors which result from the insertion and deletion of s on either side of a The edit distance or Levenshtein metric and the Lee metric arise naturally in the context of synchronization errors In recording systems using partialresp onse with some form of sequence detection ex emplied by the PRML system describ ed in Chapter the maximumlikelihood detector tends to generate burst errors whose sp ecic characteristics can b e determined from the error events asso ciated with the underlying trellis structure We will briey survey various trellis co ded mo dulation approaches for PRML that yield co des whichcombine GI constraints with enhanced minimum Euclidean distance In practice constrained co des must limit error propagation Slidingblo ck deco ders of the most frequently used d k RLL co des and PRML GI co des typically will propagate a single detector error into a burst of length no more than eight bits For example the maximum error propagation of the industry standard RLL and RLL co des are four bits and ve bits resp ectively and the PRML co de limits errors to a single byte The conv entional practice in digital recording devices is to detect and correct such errors by use of an outer errorcorrecting co de such as a Fire co de interleaved ReedSolomon co de or a mo dication of such acode Gilb ertVarshamovtyp e lower b ounds Classical bound for the Hamming metric There are several error metrics that arise in the context of digital recording using constrained sequences For substitutiontyp e errors and bitshift errors p ossibly propagated into burst errors by the mo dulation deco der it is natural to consider errorcorrecting co des based up on the Hamming metric It is therefore of interest to investigate Hamming distance prop erties of constrained sequences The Gilb ertVarshamov b ound provides for unconstrained sequences over a nite alphab et alower b ound on the size of co des with presp ecied minimum Hamming distance In this CHAPTER ERRORCORRECTING CONSTRAINED CODING section we present bounds of the Gilb ertVarshamov typ e and apply them to the class of runlengthlimited binary sequences Let denote a nite alphab et of size jj and denote the Hamming distance b etween two q q q w r be the Hamming words w w by w w For a word w let B Hamming q sphere of radius r in centered at w that is q w w r g w r fw B Hamming q q Let V w r be the cardinality or volume of the sphere B w r This quantity is P q r i jj indep endent of the center word w so we will use the shorthand nota i i q tion V r The Gilb ertVarshamov b ound provides a lower bound on the achievable cardinalit y M q of a subset of with minimum Hamming distance at least d We will refer to such a subset q as a Mdco de q Theorem There exists a Mdcode with q jj M q V d For future reference we recall that the pro of of this b ound is obtained by iteratively selecting the l th co deword w in the co de from the complement of the union of Hamming l l q q spheres of radius q centered at the previously selected co dewords B w d i i q q Continuing this pro cedure until the union of spheres exhausts an Mdco de is obtained whose size M satises the claimed inequality Let dq denote the relative minimum distance and let H z log log logz for z be a z ary generalization of the entropy function The Gilb ertVarshamov b ound in terms of the rate R of the resulting co de can b e expressed as q log M log V d R log jj log jjH jj q q for the last inequalitywe refer the reader to Berl pp Hammingmetric b ound for constrained systems Any generalization of the Gilb ertVarshamov bound to a constrained system S must take q into account that the volumes of Hamming spheres in S are not necessarily indep endent of the sp ecied centers Before deriving such b ounds we require a few more denitions Let CHAPTER ERRORCORRECTING CONSTRAINED CODING q X denote an arbitrary subset of For a word w X we dene the Hamming sphere of radius r in X by q B w r B w r X X The maximum volume of the spheres of radius r in X is V r maxjB w r j Xmax X w X and the average volume of spheres of radius r in X is given by X r V jB w r j X X jX j w X We also dene the set B r of pairs w w ofwords in X at distance no greater than r X B r fw w w w r g X Hamming q Note that jB r j V r jX j Finallywe dene an X M dco de to b e a Mdco de X X that is a subset of X tforward application of the Gilb ertVarshamov construction yields the following A straigh result q Lemma Let X be a subset of and d be a positive integer Then there exists an X M dcode with jX j M V d Xmax The following generalization of the Gilb ertVarshamov bound rst proved by Kolesnik and Krachkovsky KolK is the basis for the more rened bounds derived later in the section It provides a b ound based up on the average volume of spheres rather than the maximum volume as was used in Lemma q Lemma Let X be a subset of and d be a positive integer Then there exists an ode with X M dc jX j jX j M V d jB d j X X Pro of Consider the subset X of words w X whose Hamming spheres of radius d satisfy jB w dj V d The subset X must then satisfy jX j jX j If we X X iteratively select co dewords from X following the pro cedure used in the derivation of the Gilb ertVarshamov b ound we obtainanX M dco de where jX j j jX jX j M V d V d jB d j X X X CHAPTER ERRORCORRECTING CONSTRAINED CODING as desired In general neither of the b ounds in the preceding two lemmas is strictly sup erior to the other as observed byGuandFuja GuF However using an analysis of a new co de search algorithmdubb ed the altruistic algorithm to distinguish it from the greedy algorithm that lies at the heart of the standard Gilb ertVarshamov form of b oundthey eliminated the factor of in the denominator of the b ound in Lemma This improved lower b ound stated below as Lemma is always at least as go o d as the b ound in Lemma and a strict improvement over Lemma The key element of the improved co de search algorithm is that at eachcodeword selection step the remaining p otential co deword with the largest number of remaining neighb ors at distance dorlesstakes itself out of consideration As noted in GuF a similar approach was develop ed indep endently by Ytrehus Yta who applied it to compute b ounds for runlengthlimited co des with various error detection and correction capabilities Ytb q Lemma Let X be a subset of and d be a positive integer Then there exists an X M dcode with jX j jX j M V d jB dj X X Kolesnik and Krachkovsky KolK applied Lemma to sets X consisting of words of length q in runlengthlimited and charge constrained systems Their asymptotic lower bound was based up on an

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